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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and

Uncertainty in Business

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Chapter FourNumerical Descriptive Techniques

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Measures of Central Location

Arithmetic Mean

Mean or Average

Population Mean:

Sample Mean:

xi = the data values labeled 1 through N or n

4-3

N

xN

i

i 1

n

x

x

n

i

i 1

Measures of Central Location

Mean Time Spent on the Internet

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0 7 12 5 33 14 8 0 9 22

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Measures of Central Location

Median

The middle value

Median Time Spent on the Internet

Must first sort the date into ascending order

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0 0 5 7 8 9 12 14 22 33

Measures of Central Location

Median Long-Distance Telephone Bill

Recall our Long-Distance Telephone Bill

Data from Data File Xm03-01.xls

Interpretation: half the telephone bills are

below 26.905 and half are above 26.905

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Measures of Central Location

Mode

Observation that occurs most frequently

Mode Time Spent on the Internet

Best to sort the date into ascending order

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0 0 5 7 8 9 12 14 22 33

Measures of Central Location

Mode Long-Distance Telephone Bill

Recall our Long-Distance Telephone Bill

Data from Data File Xm03-01.xls

Two issues with the mode:

may not be “central”

may not be unique4-9

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Measures of Variability

Range

Largest observation – Smallest observation

Advantage: simple

Disadvantage: simple

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Measures of Variability

Range

Set 1

Set 2

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4 4 4 4 4 50

4 8 15 24 39 50

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Measures of Variability

Variance

Population Variance:

Sample Variance:

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N

xN

i

i

1

2

2

1

1

2

2

n

xx

s

n

i

i

Why dividing by n-1.pdf

Measures of Variability

Calculating Sample Variance

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8 4 9 11 3

ix xxi 2xxi

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Measures of Variability

Standard Deviation

Population St. Deviation:

Sample St. Deviation:

Recall our Long-Distance Telephone Bill

Data from Data File Xm03-01.xls4-17

2

2ss

Measures of Variability

Comparing Two Data Sets

Let’s explore the interpretation of variance

using Excel and Data File Xm04-08.xls

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Empirical Rule

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Empirical Rule - example

A histogram of returns on an investment is bell-shaped and has a mean of 10% and a s = 8%.

How is the distribution of the returns?

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Chebysheff’s Theorem

The proportion of observations

in any sample or population that lie

within k standard deviations of the meanis at least

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1for 1

12

kk

Chebysheff’s Theorem –example

Salaries of a computer store are positively

skewed. Mean =$28,000, std.dev=$3,000.

What can you say about these salaries?

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Measures of Variability

Coefficient of Variation

ratio of standard deviation to mean

Population Coeff. Of Variation:

Sample Coeff. Of Variation:

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CV

x

scv

Measures of Relative Standing

Percentile

the Pth percentile is the value for which

P% are less than that value and

(100-P)% are greater than that value

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Measures of Relative Standing

Quartiles

Values that divide our data set into fourths

Q1 = 25th percentile

Q2 = 50th percentile

Q3 = 75th percentile

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Measures of Relative Standing

The location of a percentile

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100

1P

nLP

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Measures of Relative Standing

Calculate the 25th, 50th and 75th Percentiles

for Time Spent on the Internet(Quartiles)

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0 0 5 7 8 9 12 14 22 33

Measures of Relative Standing

Interquartile Range

another measure of variability

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13 QQIQR

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Box Plot

Graph of five statistics

Minimum and maximum data values

1st, 2nd and 3rd Quartiles

Determine outliers using IQR

1.5 times IQR less than Q1

1.5 times IQR larger than Q3

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Measures of Linear Relationship

Covariance

Population Covariance:

Sample Covariance:

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N

yxN

i

yixi

xy

1

11

n

yyxx

s

n

i

ii

xy

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Measures of Linear Relationship

Calculating Covariance

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x y

4 24

9 27

2 18

mean 5 23

xxi yy i yyxx ii

Measures of Linear Relationship

Coefficient of Correlation

Population Correlation:

Sample Correlation:

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yx

xy

xy

yx

xy

xyss

sr

11 11 r

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Measures of Linear Relationship

Least Squares Method

Simple Linear Regression(simple => two variables only)

Two variables

Independent Variable

Dependent Variable

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Measures of Linear Relationship

Which variable is which?

Example:

Salary vs. Grocery Bill

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Measures of Linear Relationship

Let’s explore the Least Squares Method

using Data File Xm04-17.xls

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Measures of Linear Relationship

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Regression Line

+e

-e

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Measures of Linear Relationship

Least Squares Method

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xbby 10ˆ

bmxy

Measures of Linear Relationship

Least Squares Method

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21

x

xy

s

sb

xbyb 10